ON THE MOTION OF PENDULUMS. 131 second, up to the order i, as well as ^r (#, t) itself, vanish when # = 0. Substituting now in the second equation (8) the expression for v given by (184), we see that the equation is satisfied provided 3 +«•«*-«. f These equations give <£ <X, t) = x 00 e"^ t « 0 = 0- 0*0 where ^, a- denote two new arbitrary functions. Substituting in (184), and then passing to the first of the equations of condition, we get 0 = x(aO + er(aO, whence cr (x) = — % (%) and o r«5 /*oo 0 = - I cos a (# + x) e ~ ^ % (xf) dx da 71" J 0 J 0 I /• = -7=- VTT/JitJ i? (185). The second of the equations of condition requires that V= ~jL= r e""fe x 00 *»' = -T- f e~*2 % (25 V/z7^) ds. W/tfJo V^A ^V ^ Since the second member of this equation must be independent of t, we get x 00 = a constant, and this constant must be equal to F, since 2 r .# , -j- I e * Substituting in (185) we get 7 r fl =-------. € VTTLi't J 0 .(186). For the object of the present investigation nothing is required but the value of ^~ for x = 0, which we may denote by ( -y- ] . Wo G>&' \vix/ Q get from (186) dxj .(187). Now suppose the plane to be moved in any manner, so that its velocity at the end of the time t is equal to f(t). We may evidently obtain the result for this case by writing/' (t')dt' for F, 9—2